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Math Help - inequalities

  1. #1
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    inequalities

    given that x+y=1, prove that 1/x + 1/y >= 4

    Appreciate any help/hints!
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  2. #2
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    Quote Originally Posted by ose90 View Post

    given that x+y=1, prove that 1/x + 1/y >= 4

    Appreciate any help/hints!
    you also need to have x > 0 and y > 0. first note that from (t-1)^2 \geq 0, we get: t + \frac{1}{t} \geq 2, for any t > 0. thus: \frac{1}{x} + \frac{1}{y} = (x+y) \left(\frac{1}{x} + \frac{1}{y} \right)=2 + \frac{x}{y} + \frac{y}{x} \geq 4.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    you also need to have x > 0 and y > 0. first note that from (t-1)^2 \geq 0, we get: t + \frac{1}{t} \geq 2, for any t > 0. thus: \frac{1}{x} + \frac{1}{y} = (x+y) \left(\frac{1}{x} + \frac{1}{y} \right)=2 + \frac{x}{y} + \frac{y}{x} \geq 4.
    Thanks alot!
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